Finite Group Databasebeta

← #13154237

Subgroups — #13154237

Order 1728.

1080 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#126C211yesmaximal, characteristic, complemented
2#11721754C211yesmaximal, characteristic, complemented
3#11721753C211yesmaximal, characteristic, complemented
4#11721752C211yesmaximal, characteristic, complemented
5#11721750C211yesmaximal, characteristic, complemented
6#11721748C211yesmaximal, characteristic, complemented
7#11721744C211yesmaximal, characteristic, complemented
8#11721732C211yesmaximal, characteristic, complemented
9#11721731C211yesmaximal, characteristic, complemented
10#11721731C211yesmaximal, characteristic, complemented
11#11721730C211yesmaximal, characteristic
12#11721727C211yesmaximal, characteristic, complemented
13#11721724C211yesmaximal, characteristic, complemented
14#11721712C211yesmaximal, characteristic, complemented
15C4×#6226C211yesmaximal, characteristic, complemented
16#10596181—31nomaximal
17#10589812—33nomaximal
18C2×#6233C2211yescharacteristic
19C2×#6233C2211yescharacteristic
20C22×C3⋊Dic9C2211yescharacteristic
21C2×C4⋊(C3⋊D9)C2211yescharacteristic
22C2×C4×C3⋊D9C2211yescharacteristic
23C2×C4×C3⋊D9C2211yescharacteristic
24C2×#6226C2211yescharacteristic
25C2×#6226C2211yescharacteristic
26C2×#6226C2211yescharacteristic
27C3×C9×C4⋊C4C2211yescharacteristic
28C3×C9×C4⋊C4C2211yescharacteristic
29C3×C9×C4⋊C4C2211yescharacteristic
30C3×C9×C22⋊C4C2211yescharacteristic
31C3×C9×C22⋊C4C2211yescharacteristic
32C3×C9×C22⋊C4C2211yescharacteristic
33C3×C42×C9C2211yescharacteristic
34#88551C2211yescharacteristic
35#88536C2211yescharacteristic
36#88536C2211yescharacteristic
37#88536C2211yescharacteristic
38#88536C2211yescharacteristic
39#88536C2211yescharacteristic
40#88536C2211yescharacteristic
41#88536C2211yescharacteristic
42#88536C2211yescharacteristic
43C4⋊(C3⋊Dic9)C2211yescharacteristic
44C4⋊(C3⋊Dic9)C2211yescharacteristic
45C4⋊(C3⋊Dic9)C2211yescharacteristic
46#88534C2211yescharacteristic
47#88534C2211yescharacteristic
48#88534C2211yescharacteristic
49#88534C2211yescharacteristic
50C4×C3⋊Dic9C2211yescharacteristic
51C4×C3⋊Dic9C2211yescharacteristic
52C4×C3⋊Dic9C2211yescharacteristic
53C32×#126D311yescharacteristic
54#64107—31no—
55#64106—31no—
56#64105—31no—
57#64103—31no—
58#64101—31no—
59#64097—31no—
60#64095—31no—
61#64094—31no—
62#64094—31no—
63#64093—31no—
64#64090—31no—
65#64087—31no—
66#64084—31no—
67C4×#634—31no—
68C9×#126D313yescomplemented
69#63460—33no—
70#63459—33no—
71#63458—33no—
72#63456—33no—
73#63454—33no—
74#63450—33no—
75#63448—33no—
76#63447—33no—
77#63447—33no—
78#63446—33no—
79#611⋊C4—33no—
80#63440—33no—
81#63437—33no—
82C4×Dic18—33no—
83C23×C3×C9C2311yescharacteristic
84C22×C3⋊D9C2311yescharacteristic
85C22×C3⋊D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3×C4×C9C2311yescharacteristic
90C2×C3×C4×C9C2311yescharacteristic
91C2×C3×C4×C9C2311yescharacteristic
92#6233—22no—
93#6233—41no—
94C2×C3⋊Dic9—41no—
95C2×C3⋊Dic9C2311yescharacteristic
96C2×C3⋊Dic9C2311yescharacteristic
97C2×C3⋊Dic9C2311yescharacteristic
98C2×C3⋊Dic9C2311yescharacteristic
99C2×C3⋊Dic9C2311yescharacteristic
100C2×C3⋊Dic9C2311yescharacteristic
101C4⋊(C3⋊D9)—41no—
102C4×C3⋊D9—22no—
103C4×C3⋊D9—41no—
104#6226—22no—
105#6226—41no—
106#6226—41no—
107#5710—93no—
108#5710—91no—
109C2×#638—31no—
110C2×#638—31no—
111C22×C3⋊Dic3—31no—
112C2×C4⋊(C3⋊D3)—31no—
113C2×C4×C3⋊D3—31no—
114C2×C4×C3⋊D3—31no—
115C2×#634—31no—
116C2×#634—31no—
117C2×#634—31no—
118C32×C4⋊C4C2×D311yescharacteristic
119C32×C4⋊C4C2×D311yescharacteristic
120C32×C4⋊C4C2×D311yescharacteristic
121C32×C22⋊C4C2×D311yescharacteristic
122C32×C22⋊C4C2×D311yescharacteristic
123C32×C22⋊C4C2×D311yescharacteristic
124C32×C42C2×D311yescharacteristic
125#3757—31no—
126#3752—31no—
127#3752—31no—
128#3752—31no—
129#3752—31no—
130#3752—31no—
131#3752—31no—
132#3752—31no—
133#3752—31no—
134C4⋊(C3⋊Dic3)—31no—
135C4⋊(C3⋊Dic3)—31no—
136C4⋊(C3⋊Dic3)—31no—
137#3750—31no—
138#3750—31no—
139#3750—31no—
140#3750—31no—
141C4×C3⋊Dic3—31no—
142C4×C3⋊Dic3—31no—
143C4×C3⋊Dic3—31no—
144C2×#611—33no—
145C2×#611—33no—
146C22×Dic9—33no—
147C2×D36—33no—
148C2×C4×D9—33no—
149C2×C4×D9—33no—
150C2×Dic18—33no—
151C2×Dic18—33no—
152C2×Dic18—33no—
153C9×C4⋊C4C2×D313yes—
154C9×C4⋊C4C2×D313yes—
155C9×C4⋊C4C2×D313yes—
156C9×C22⋊C4C2×D313yes—
157C9×C22⋊C4C2×D313yes—
158C9×C22⋊C4C2×D313yes—
159C42×C9C2×D313yes—
160#3676—33no—
161(C2×D9)⋊C4—33no—
162(C2×D9)⋊C4—33no—
163(C2×D9)⋊C4—33no—
164(C2×D9)⋊C4—33no—
165(C2×D9)⋊C4—33no—
166(C2×D9)⋊C4—33no—
167(C2×D9)⋊C4—33no—
168(C2×D9)⋊C4—33no—
169C4⋊Dic9—33no—
170C4⋊Dic9—33no—
171C4⋊Dic9—33no—
172Dic9⋊C4—33no—
173Dic9⋊C4—33no—
174Dic9⋊C4—33no—
175Dic9⋊C4—33no—
176C4×Dic9—33no—
177C4×Dic9—33no—
178C4×Dic9—33no—
179C22×C3×C9—21no—
180C22×C3×C9—21no—
181C22×C3×C9—21no—
182C22×C3×C9C2411yescharacteristic
183C2×C3⋊D9—21no—
184C2×C3⋊D9—21no—
185C2×C3⋊D9—21no—
186C2×C3⋊D9—21no—
187C2×C3⋊D9—21no—
188C2×C3⋊D9—21no—
189C3×C4×C9—21no—
190C3×C4×C9—21no—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3×C4×C9—21no—
194C3×C4×C9—21no—
195C3⋊Dic9—21no—
196C3⋊Dic9—21no—
197C3⋊Dic9—21no—
198C3⋊Dic9—21no—
199C3⋊Dic9—21no—
200C3⋊Dic9#4112yes—
201C3×#126D913yescomplemented
202C3×#126C3⋊D311yescharacteristic
203#898—93no—
204#898—91no—
205#897—93no—
206#897—91no—
207#896—93no—
208#896—91no—
209#894—93no—
210#894—91no—
211#892—93no—
212#892—91no—
213#888—93no—
214#888—91no—
215#886—93no—
216#886—91no—
217#885—93no—
218#885—93no—
219#885—91no—
220#885—91no—
221#884—93no—
222#884—91no—
223#67⋊C4—93no—
224#67⋊C4—91no—
225#878—93no—
226#878—91no—
227#875—93no—
228#875—91no—
229C4×Dic6—93no—
230C4×Dic6—91no—
231C23×C32C22×D311yescharacteristic
232C22×C3⋊D3—31no—
233C22×C3⋊D3—31no—
234C2×C32×C4C22×D311yescharacteristic
235C2×C32×C4C22×D311yescharacteristic
236C2×C32×C4C22×D311yescharacteristic
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239C2×C32×C4C22×D311yescharacteristic
240#638—62no—
241#638—121no—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—31no—
244C2×C3⋊Dic3—31no—
245C2×C3⋊Dic3—31no—
246C2×C3⋊Dic3—31no—
247C2×C3⋊Dic3—31no—
248C2×C3⋊Dic3—121no—
249C4⋊(C3⋊D3)—121no—
250C4×C3⋊D3—62no—
251C4×C3⋊D3—121no—
252#634—62no—
253#634—121no—
254#634—121no—
255C23×C9C22×D313yes—
256C22×D9—33no—
257C22×D9—33no—
258C2×C4×C9C22×D313yes—
259C2×C4×C9C22×D313yes—
260C2×C4×C9C22×D313yes—
261C2×C4×C9C22×D313yes—
262C2×C4×C9C22×D313yes—
263C2×C4×C9C22×D313yes—
264#611—66no—
265#611—123no—
266C2×Dic9—33no—
267C2×Dic9—33no—
268C2×Dic9—33no—
269C2×Dic9—33no—
270C2×Dic9—33no—
271C2×Dic9—33no—
272C2×Dic9—123no—
273D36—123no—
274C4×D9—66no—
275C4×D9—123no—
276Dic18—66no—
277Dic18—123no—
278Dic18—123no—
279#542—271noSylow
280C2×C3×C9—41no—
281C2×C3×C9C2×#4111yescharacteristic
282C2×C3×C9H(5,𝔽2)11yescharacteristic
283C2×C3×C92−1+411yescharacteristic
284C3⋊D9—41no—
285C3⋊D9—41no—
286C2×#67—93no—
287C2×#67—91no—
288C2×#67—93no—
289C2×#67—91no—
290C22×Dic3—93no—
291C22×Dic3—91no—
292C2×D12—93no—
293C2×D12—91no—
294C2×C4×D3—93no—
295C2×C4×D3—91no—
296C2×C4×D3—93no—
297C2×C4×D3—91no—
298C2×Dic6—93no—
299C2×Dic6—91no—
300C2×Dic6—93no—
301C2×Dic6—93no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C3×C4⋊C4C2×D913yes—
305C3×C4⋊C4C2×D913yes—
306C3×C4⋊C4C2×D913yes—
307C3×C4⋊C4C2×C3⋊D311yescharacteristic
308C3×C4⋊C4C2×C3⋊D311yescharacteristic
309C3×C4⋊C4C2×C3⋊D311yescharacteristic
310C3×C22⋊C4C2×D913yes—
311C3×C22⋊C4C2×D913yes—
312C3×C22⋊C4C2×D913yes—
313C3×C22⋊C4C2×C3⋊D311yescharacteristic
314C3×C22⋊C4C2×C3⋊D311yescharacteristic
315C3×C22⋊C4C2×C3⋊D311yescharacteristic
316C3×C42C2×D913yes—
317C3×C42C2×C3⋊D311yescharacteristic
318#217—93no—
319#217—91no—
320(C2×D3)⋊C4—93no—
321(C2×D3)⋊C4—93no—
322(C2×D3)⋊C4—93no—
323(C2×D3)⋊C4—93no—
324(C2×D3)⋊C4—93no—
325(C2×D3)⋊C4—93no—
326(C2×D3)⋊C4—93no—
327(C2×D3)⋊C4—93no—
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4—91no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—91no—
332(C2×D3)⋊C4—91no—
333(C2×D3)⋊C4—91no—
334(C2×D3)⋊C4—91no—
335(C2×D3)⋊C4—91no—
336C4⋊Dic3—93no—
337C4⋊Dic3—93no—
338C4⋊Dic3—93no—
339C4⋊Dic3—91no—
340C4⋊Dic3—91no—
341C4⋊Dic3—91no—
342Dic3⋊C4—93no—
343Dic3⋊C4—93no—
344Dic3⋊C4—93no—
345Dic3⋊C4—93no—
346Dic3⋊C4—91no—
347Dic3⋊C4—91no—
348Dic3⋊C4—91no—
349Dic3⋊C4—91no—
350C4×Dic3—93no—
351C4×Dic3—93no—
352C4×Dic3—93no—
353C4×Dic3—91no—
354C4×Dic3—91no—
355C4×Dic3—91no—
356C22×C32—21no—
357C22×C32—21no—
358C22×C32—21no—
359C22×C32C23×D311yescharacteristic
360C2×C3⋊D3—61no—
361C2×C3⋊D3—61no—
362C2×C3⋊D3—61no—
363C2×C3⋊D3—61no—
364C2×C3⋊D3—61no—
365C2×C3⋊D3—61no—
366C32×C4—21no—
367C32×C4—21no—
368C32×C4—21no—
369C32×C4—21no—
370C32×C4—21no—
371C32×C4—21no—
372C3⋊Dic3—32no—
373C3⋊Dic3—61no—
374C3⋊Dic3—61no—
375C3⋊Dic3—61no—
376C3⋊Dic3—61no—
377C3⋊Dic3—61no—
378C22×C9—23no—
379C22×C9—23no—
380C22×C9—23no—
381C22×C9C23×D313yes—
382C2×D9—63no—
383C2×D9—63no—
384C2×D9—63no—
385C2×D9—63no—
386C2×D9—63no—
387C2×D9—63no—
388C4×C9—23no—
389C4×C9—23no—
390C4×C9—23no—
391C4×C9—23no—
392C4×C9—23no—
393C4×C9—23no—
394Dic9—36no—
395Dic9—63no—
396Dic9—63no—
397Dic9—63no—
398Dic9—63no—
399Dic9—63no—
400C4⋊Q8—271no—
401#126—271no—
402#126C3⋊D911yescharacteristic, complemented
403#124—271no—
404#124—271no—
405#124—271no—
406#123—271no—
407#123—271no—
408C22⋊Q8—271no—
409C22⋊Q8—271no—
410C22⋊Q8—271no—
411#121—271no—
412C4×Q8—271no—
413C4×D4—271no—
414#117—271no—
415C3×C9#54211yescharacteristic, complemented, Sylow
416C23×C3C22×D913yes—
417C23×C3C22×C3⋊D311yescharacteristic
418C22×D3—91no—
419C22×D3—91no—
420C22×D3—93no—
421C22×D3—93no—
422C2×C3×C4C22×D913yes—
423C2×C3×C4C22×D913yes—
424C2×C3×C4C22×D913yes—
425C2×C3×C4C22×D913yes—
426C2×C3×C4C22×D913yes—
427C2×C3×C4C22×D913yes—
428C2×C3×C4C22×C3⋊D311yescharacteristic
429C2×C3×C4C22×C3⋊D311yescharacteristic
430C2×C3×C4C22×C3⋊D311yescharacteristic
431C2×C3×C4C22×C3⋊D311yescharacteristic
432C2×C3×C4C22×C3⋊D311yescharacteristic
433C2×C3×C4C22×C3⋊D311yescharacteristic
434#67—182no—
435#67—186no—
436#67—361no—
437#67—363no—
438C2×Dic3—91no—
439C2×Dic3—93no—
440C2×Dic3—91no—
441C2×Dic3—91no—
442C2×Dic3—91no—
443C2×Dic3—91no—
444C2×Dic3—91no—
445C2×Dic3—93no—
446C2×Dic3—93no—
447C2×Dic3—93no—
448C2×Dic3—93no—
449C2×Dic3—93no—
450C2×Dic3—361no—
451C2×Dic3—363no—
452D12—361no—
453D12—363no—
454C4×D3—182no—
455C4×D3—186no—
456C4×D3—361no—
457C4×D3—363no—
458Dic6—182no—
459Dic6—186no—
460Dic6—361no—
461Dic6—361no—
462Dic6—363no—
463Dic6—363no—
464C2×C32—41no—
465C2×C32D3×#4111yescharacteristic
466C2×C32#100911yescharacteristic
467C2×C32#101011yescharacteristic
468C3⋊D3—121no—
469C3⋊D3—121no—
470C2×C9—43no—
471C2×C9D3×#4113yes—
472C2×C9#100913yes—
473C2×C9#101013yes—
474D9—123no—
475D9—123no—
476C2×Q8—271no—
477C2×Q8—271no—
478C2×Q8—271no—
479C2×D4—271no—
480C2×D4—271no—
481C2×D4—271no—
482C22×C4—271no—
483C22×C4—271no—
484C22×C4—271no—
485C4⋊C4—271no—
486C4⋊C4—271no—
487C4⋊C4—271no—
488C4⋊C4—271no—
489C4⋊C4—271no—
490C4⋊C4—271no—
491C4⋊C4—271no—
492C4⋊C4C2×C3⋊D911yescharacteristic
493C4⋊C4C2×C3⋊D911yescharacteristic
494C4⋊C4C2×C3⋊D911yescharacteristic
495C22⋊C4—271no—
496C22⋊C4—271no—
497C22⋊C4—271no—
498C22⋊C4—271no—
499C22⋊C4—271no—
500C22⋊C4—271no—
501C22⋊C4—271no—
502C22⋊C4—271no—
503C22⋊C4—271no—
504C22⋊C4C2×C3⋊D911yescharacteristic
505C22⋊C4C2×C3⋊D911yescharacteristic
506C22⋊C4C2×C3⋊D911yescharacteristic
507C42—271no—
508C42—271no—
509C42—271no—
510C42C2×C3⋊D911yescharacteristic
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—23no—
515C22×C3—23no—
516C22×C3—23no—
517C22×C3C23×D913yes—
518C22×C3C23×C3⋊D311yescharacteristic
519C2×D3—181no—
520C2×D3—181no—
521C2×D3—181no—
522C2×D3—181no—
523C2×D3—181no—
524C2×D3—181no—
525C2×D3—183no—
526C2×D3—183no—
527C2×D3—183no—
528C2×D3—183no—
529C2×D3—183no—
530C2×D3—183no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—23no—
538C3×C4—23no—
539C3×C4—23no—
540C3×C4—23no—
541C3×C4—23no—
542C3×C4—23no—
543Dic3—92no—
544Dic3—96no—
545Dic3—181no—
546Dic3—181no—
547Dic3—181no—
548Dic3—181no—
549Dic3—181no—
550Dic3—183no—
551Dic3—183no—
552Dic3—183no—
553Dic3—183no—
554Dic3—183no—
555C32#571011yescharacteristic
556C9#571013yescomplemented
557C23—271no—
558C23—271no—
559C23C22×C3⋊D911yescharacteristic
560Q8—542no—
561Q8—1081no—
562Q8—1081no—
563D4—542no—
564D4—1081no—
565D4—1081no—
566C2×C4—271no—
567C2×C4—271no—
568C2×C4—271no—
569C2×C4—271no—
570C2×C4—271no—
571C2×C4—271no—
572C2×C4—542no—
573C2×C4—1081no—
574C2×C4—1081no—
575C2×C4C22×C3⋊D911yescharacteristic
576C2×C4C22×C3⋊D911yescharacteristic
577C2×C4C22×C3⋊D911yescharacteristic
578C2×C4C22×C3⋊D911yescharacteristic
579C2×C4C22×C3⋊D911yescharacteristic
580C2×C4C22×C3⋊D911yescharacteristic
581C2×C3—41no—
582C2×C3—43no—
583C2×C3#41×D913yes—
584C2×C3#6371513yes—
585C2×C3#6371613yes—
586C2×C3#41×C3⋊D311yescharacteristic
587C2×C3#6436611yescharacteristic
588C2×C3#6436711yescharacteristic
589D3—361no—
590D3—361no—
591D3—363no—
592D3—363no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—541no—
597C22—541no—
598C22—541no—
599C22—541no—
600C22—541no—
601C22—541no—
602C22C23×C3⋊D911yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—272no—
610C4—541no—
611C4—541no—
612C4—541no—
613C4—541no—
614C4—541no—
615C3#1058981213yescomplemented
616C3#1059618111yescharacteristic
617C2—41no—
618C2—1081no—
619C2—1081no—
620C2#41×C3⋊D911yescharacteristic
621C2#1172267211yescharacteristic
622C2#1172267311yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#126C2
socleC22×C32C23×D3
prime radical#11721730C2
hypercenter#126C3⋊D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #126

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×C9×#126C2semidirect
2. #11721754C2semidirect
3. #11721753C2semidirect
4. #11721752C2semidirect
5. #11721750C2semidirect
6. #11721748C2semidirect
7. #11721744C2semidirect
8. #11721732C2semidirect
9. #11721731C2semidirect
10. #11721731C2semidirect
12. #11721727C2semidirect
13. #11721724C2semidirect
14. #11721712C2semidirect
15. C4×#6226C2semidirect
68. C9×#126D3semidirect
201. C3×#126D9semidirect
402. #126C3⋊D9semidirect
415. C3×C9#542semidirect
556. C9#5710semidirect
615. C3#10589812semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C9×#1262
#117217542
#117217532
#117217522
#117217502
#117217482
#117217442
#117217322
#117217312
#117217302
#117217272
#117217242
#117217122
C4×#62262
#105961813
#105898123